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For a Sato-Tate group $G$ of degree $d$ and weight $w$ a Hodge circle is a continuous homomorphism \[ h\colon \mathrm{U}(1)\to G^0 \] for which there exist nonnegative integers $h^{p,q}$ satisfying $\sum h^{p,q}=d$ with $p,q$ varying over integers in $[0,w]$ such that

  • $h(u)$ has eigenvalues $u^{p-q}$ with multiplicity $h^{p,q}$ for all $u\in \mathrm{U}(1)$;
  • the conjugates of the image of $h$ generate a dense subgroup of $G^0$.

When $G$ is the Sato-Tate group of a motive $X$ of weight $w$, the $h^{p,q}$ are the Hodge numbers of $X$. For a $d$-dimensional abelian variety $X$ we necessarily have $h^{0,1}=h^{1,0}=d$.

Knowl status:
  • Review status: beta
  • Last edited by Andrew Sutherland on 2021-01-16 14:47:48
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